Dynamic algorithms for k-center on graphs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917561758646272 |
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| author | Cruciani, Emilio Forster, Sebastian Goranci, Gramoz Nazari, Yasamin Skarlatos, Antonis |
| author_facet | Cruciani, Emilio Forster, Sebastian Goranci, Gramoz Nazari, Yasamin Skarlatos, Antonis |
| contents | In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem.
While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental $(2+ε)$-approximation algorithm and a randomized incremental $(4+ε)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+ε)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art fully dynamic $(1+ε)$-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+ε)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15557 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamic algorithms for k-center on graphs Cruciani, Emilio Forster, Sebastian Goranci, Gramoz Nazari, Yasamin Skarlatos, Antonis Data Structures and Algorithms Machine Learning In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental $(2+ε)$-approximation algorithm and a randomized incremental $(4+ε)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+ε)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art fully dynamic $(1+ε)$-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+ε)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances. |
| title | Dynamic algorithms for k-center on graphs |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2307.15557 |